Together with Daniel Matthes and Giuseppe Savaré, we study the quantum drift-diffusion, or Derrida–Lebowitz–Speer–Spohn (DLSS), equation for a nonnegative density $\varrho$ on a bounded convex domain with Neumann boundary conditions, in the square-root variable $u=\sqrt{\varrho }$. We show that the DLSS operator, defined and monotone on smooth strictly positive functions, admits a unique maximal monotone extension in $L^2(\mathrm{\Omega })$, explicitly given by the minimal (defect-free) operator plus the normal cone of the positivity constraint. The generated semigroup, which contracts the Hellinger distance between the densities, thus yields a canonical solution — existing, unique, and stable for every nonnegative $L^2$ initial datum and in every space dimension — independent of any approximation scheme: it is in fact the unique contraction semigroup extending the classical evolutions that emanate from smooth, uniformly positive data. The implicit Euler scheme converges to it, and $\sqrt{u}\in L_{\mathrm{loc}}^2(H^2)$ along the flow. When the datum belongs to the domain of the operator, the solution is strong and satisfies the equation pointwise, with no reaction term created on the vacuum $\{u=0\}$. We characterize the trajectories in several equivalent ways — as Bénilan integral solutions and through one-sided weak formulations — prove the maximality of the operator also in the $H^2$–$H^{-2}$ duality and, in dimension $d\le 3$, identify the flow with the weak solutions in the uniqueness class of Fischer. A second-order estimate of independent interest underlies the construction: on a convex domain with Neumann conditions the dissipation ${\int }_{\mathrm{\Omega }}{(\mathrm{\Delta }u)}^2/u𝑑x$ is finite exactly when $\sqrt{u}\in H^2(\mathrm{\Omega })$, and it then controls the full Hessian of $\sqrt{u}$, in every dimension.
arXiv:2608.16792v1